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Heights of Cylinders Given Surface Area

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What if you know the surface area of a cylinder, but don't know the height? Look at this dilemma.

Candice is going to use modge podge to decorate the outside of a cylindrical canister. She wants to make it into a decorative pencil holder as a gift for her Mom. Modge podge is a glue - like substance that helps to adhere tissue paper or other decorative pieces of paper to an object.

Candice's canister has a radius of 3 inches and has a surface area of 207.3 \ in^2 .

What is the height of the canister?

Use this Concept to learn about how to figure out the height of a cylinder given the radius and surface area.

Guidance

Sometimes we may be given the surface area of a cylinder and need to find its height. We simply fill the information we know into the surface area formula. This time, instead of solving for SA , we solve for h , the height of the cylinder.

Let’s look at how we can apply this information.

The surface area of a cylinder with a radius of 3 inches is 78 \pi square inches. What is the height of the cylinder?

We know that the radius of the cylinder is 3 inches. We also know that the surface area of the cylinder is 78 \pi square inches. Sometimes a problem will include pi in the amount because it is more precise. We’ll see that this also makes our calculations easier. All we have to do is put 78 \pi into the formula in place of SA . Then we can use the formula to solve for the height, h .

SA & = 2 \pi r^2 + 2 \pi rh\\78 \pi & = 2 \pi (3^2) + 2 \pi (3) h\\78 \pi & = 2 \pi (9) + 6 \pi h\\78 \pi & = 18 \pi + 6 \pi h\\78 \pi - 18 \pi & = 6 \pi h \qquad \qquad \qquad \text{Subtract} \ 18 \pi \ \text{from both sides.}\\60 \pi & = 6 \pi h\\60 \pi \div 6 \pi & = h \qquad \qquad \quad \qquad \text{Divide both sides by} \ 6 \pi.\\10 \ in. & = h

We used the formula to find that a cylinder with a radius of 3 inches and a surface area of 78 \pi has a height of 10 inches.

We can check our work by putting the height into the formula and solving for surface area.

SA & = 2 \pi (3^2) + 2 \pi (3) (10)\\SA & = 2 \pi (9) + 2 \pi (30)\\SA & = 18 \pi + 60 \pi\\SA & = 78 \pi

This is the surface area given in the problem, so our answer is correct. Let’s look at another example.

Now try a few of these on your own.

Find the height in each problem.

Example A

SA = 48 \pi, \ r = 3 \ in

Solution: 2 \ in

Example B

SA = 60 \pi, \ r = 2 \ in

Solution: 5 \ in

Example C

SA = 80 \pi, \ r = 4 \ m

Solution: 2 \ m

Here is the original problem once again.

Candice is going to use modge podge to decorate the outside of a cylindrical canister. She wants to make it into a decorative pencil holder as a gift for her Mom. Modge podge is a glue - like substance that helps to adhere tissue paper or other decorative pieces of paper to an object.

Candice's canister has a radius of 3 inches and has a surface area of 207.24 \ in^2 .

What is the height of the canister?

To figure this out, let's use the formula.

SA & = 2 \pi r^2 + 2 \pi rh\\66 \pi & = 2 \pi (3^2) + 2 \pi (3) h\\66 \pi & = 2 \pi (9) + 6 \pi h\\66 \pi & = 18 \pi + 6 \pi h\\66 \pi - 18 \pi & = 6 \pi h \qquad \qquad \qquad \text{Subtract} \ 18 \pi \ \text{from both sides.}\\66 \pi & = 6 \pi h\\66 \pi \div 6 \pi & = h \qquad \qquad \quad \qquad \text{Divide both sides by} \ 6 \pi.\\8 \ in. & = h

The height of the cylinder is 8 \ in .

Vocabulary

Cylinder
a three-dimensional figure with two circular bases.
Surface Area
the measurement of the outside of a three-dimensional figure.
Net
a two-dimensional representation of a three-dimensional figure.

Guided Practice

Here is one for you to try on your own.

Mrs. Javitz bought a container of oatmeal in the shape of the cylinder. If the container has a radius of 5 cm and a surface area of 170 \pi , what is its height?

Answer

Once again, we have been given the surface area as a function of pi. That’s no problem we simply put the whole number, 170 \pi , in for SA in the formula. We also know that the container has a radius of 5 centimeters, so we can use the formula to solve for h , the height of the container.

SA & = 2 \pi r^2 + 2 \pi rh\\170 \pi & = 2 \pi (5^2) + 2 \pi (5) h\\170 \pi & = 2 \pi (25) + 10 \pi h\\170 \pi & = 50 \pi + 10 \pi h\\170 \pi - 50 \pi & = 10 \pi h \quad \qquad \qquad \text{Subtract} \ 50 \pi \ \text{from both sides.}\\120 \pi & = 10 \pi h\\120 \pi \div 10 \pi & = h \qquad \qquad \qquad \ \ \text{Divide both sides by} \ 10 \pi.\\12 \ cm & = h

The oatmeal container must have a height of 12 centimeters.

Video Review

Khan Academy, Cylinder Volume and Surface Area

Practice

Directions : Find the height of each cylinder given the surface area and one other dimension.

1. r = 5 \ in

SA = 376.8 \ in^2

2. r = 6 \ in

SA = 527.52 \ in^2

3. r = 5.5 \ in

SA = 336.8255 \ in^2

4. r = 4 \ cm

SA = 251.2 \ cm^2

5. r = 5 \ m

SA = 439.6 \ m^2

6. r = 2 \ m

SA = 87.92 \ cm^2

7. r = 2 \ cm

SA = 75.35 \ cm^2

8. d = 4 \ m

SA = 125.6 \ m^2

9. d = 8 \ cm

SA = 351.68 \ cm^2

10. d = 6 \ cm

SA = 282.6 \ cm^2

11. r = 3 \ cm

SA = 226.08 \ cm^2

12. r = 14 \ cm

SA = 2637.6 \ cm^2

13. r = 18 \ cm

SA = 4295.53 \ cm^2

14. r = 12 \ cm

SA = 1959.36 \ cm^2

15. r = 13 \ cm

SA = 3347.24 \ cm^2

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